(Mathematics) How to ask professors about their work when you only have a vague understanding of their subject area?

Question

I recently graduated from a Masters in mathematical physics. During my time I especially enjoyed pure maths courses and picked as many as I could. Since graduating I have been exploring my options and am intrigued by the subject of elliptic curves. I am trying to build a background in my spare time, with the goal of writing a short presentation of the topic, to learn by teaching and to potentially use in future applications.

If I continue enjoying the subject, I would like to apply for a PhD. I feel I should get some point of contact with potential supervisors as soon as possible, however it feels rude to do so with only a very vague understanding of their work.

My question

How do I initiate contact with academics working on elliptic curves despite being very new to the subject. Could I ask them to explain something that would be found in a textbook ? For example, the Tate-Shafarevich group seems to be quite actively researched. Could I ask them why they're interested in this group ?

Answer 1

Here's what I would recommend.

1. First, I would learn the basics of elliptic curves by consulting a standard book on elliptic curves. A good first option is the book by Silverman and Tate. You could also consult the book "The Arithmetic of Elliptic Curves" by Silverman. These books focus on the relationship to number theory. "Lectures on Elliptic Curves" by Cassels is also really good.

2. After spending some time on it, if you are interested, you could look for people who work on elliptic curves. Now you have found people who think about elliptic curves. Then what?

3. Go to their list of publications and look at some of them. You don't have to try to understand the papers completely. Just read the introductions to these papers. Try to answer for yourself: What do they do in the paper and why?

4. Then you could send them a short e-mail (5 sentences or so). You could say the following:

a. You've been reading about elliptic curves on your own, using the references [insert the books/articles you've been reading] and have been enjoying learning about them.

b. You're interested in contemporary research in this area, and you found out that they work on this topic.

c. You've looked at some of their publications and have read the introductions to their publications to get an idea of what they work on.

d. You would like to learn more. Then ask if they could recommend something to read to help them learn about current research.

This shows that you're interested in their work, and that you've done some independent reading.

(Also, roughly speaking, the reason some people are interested in the Tate-Shafarevich group is that a lot of interesting results about arithmetic aspects of elliptic curves, like their L-functions, are only true under the assumption that the Tate-Shafarevich group is finite.)

Answer 2

it feels rude to do so with only a very vague understanding of their work

Actually, it is rude, if you are not enrolled in their course , however luckily we live in the year 2022 and you can access an online course to build at least a minimum understanding of the topic, since I assume you have the time and the (intellectual) resources to attend such a course, or at least to work out the syllabus of such a course.

Additionally, you may join a workshop or a conference on the topic. If you write nicely your situation (not a student, unemployed, interested in the topic, looking for opportunities for a PhD on the topic of) to the organizers, they are likely to waive the fees to access the conference (if any).

Answer 3

How do I initiate contact with academics working on elliptic curves despite being very new to the subject. Could I ask them to explain something that would be found in a textbook?

Unlike the other individual points of view expressed here, I think this can certainly be done in a way that neither rude, spam, nor unwise.

While there are certainly "I can't be bothered" types in any academic field, there are also plenty of folks who understand that the finding various ways to explain their research, especially to people with different levels of familiarity is a valuable exercise for scholars and some would even see it as a sacred duty¹.

Enlightened scholars will also know that sometimes unexpected benefits arise from unexpected or nonstandard discussions.

I would not say that if they refuses or ignored a well-written email of this type that they were "bad" or unenlightened, but I would see their willingness to discuss with you or not as a sort of filter for the kind of person you'd like to get to know in the first place!

Okay, but what would a "well-written email of this type" be like?

It should be honest and upfront but not off-putting. Explain that you don't want a full lecture on the topic, but before you take the dive you want a little bit of guidance and/or clarification to make sure you get started on the right foot and path. Include a few specifics that make it clear you have at least some understanding of the topic and that they would not be wasting their time.

Mention a time limit, like 15 or 20 minutes. Of course if they feel it's productive they may extend, but make it clear you have every intention of keeping it short and promptly leaving if they don't offer to extend. You want to give them an opportunity to see if they can help, without fear that they have a lot to loose.

• If you get a positive response², then despite what you said, study like crazy and make at least some progress so that they might be pleasantly surprised that you were a bit modest describing your level of understanding, and so that they are further reassured that spending a little time with you has a chance of making a real difference. To that end cgb5436's answer suggests some resources to start with.
• If you get a negative response send a short, concise thank you reply and follow up on your own on any advice they might have included in their response (unless it's bad advice!)
• If you get no response then you should not pursue it further with that person at this time. While occasionally mail messages get missed or move to the spam folder, most likely they don't want to deal with the situation, and you don't want to do anything that looks troublesome or harassing or else word of you may spread in a negative way.

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• ¹in moderation.
• ²and even if you don't!

Answer 4

Unless you are enrolled in their classes or you find some mistakes in their books that you would like to report, I don't think it is a good idea to contact them (that way). In the worst case, they may mark your email as spam.

If you only want to initiate the conversation with them, please don't do it this way. It is not wise to impress them like this.

If have some doubts while reading the books. Please consider asking professors whom you know. They may provide some help.

Furthermore, do your "homework". Keep reading and studying the topic yourself. At least, you need to have a decent understanding of the topic to know if you really like it. Some topics may sound super interesting at first glance, but turn out to be terribly boring later. When you are ready, prepare the necessary documents for the application. Do some research on potential supervisors and contact them (if it is appropriate)